Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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9
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2answers
161 views

On the root of $\cos (a_1x) + \cdots + \cos (a_nx) = 0$

This is a problem I was trying to solve for a while with no succeed. Show that the equation $\cos (a_1x) + \cdots + \cos (a_nx) = 0$ has at least one solution in $[0,\frac {\pi}{a_1}]$, where $0 < ...
2
votes
2answers
154 views

Finding the complex fourier series of the function $x^2\sin(x)$ in the interval $[{-\pi}, \pi]$?

This forms part of a project I am doing and I wish to see how well complex fourier series approximates a smooth curve such as this one. After tedious integration by parts, I have attained an answer ...
1
vote
1answer
350 views

Contour Integral of Exponential

I want to show the following for $a > 0$: $$e^{-a} = \int_{0}^{\infty}{\frac{e^{-x}}{\sqrt{x}}e^{-a^{2}/(4x)}dx}.$$
2
votes
1answer
388 views

Solve Basel problem with Fourier series of $[-\pi,\pi]\to \mathbb{R}:\theta \mapsto |\theta|$

A problem from Stein/Shakarchi's Fourier Analysis: Consider the function $f:[-\pi, \pi] \to \mathbb{R}:\theta \mapsto |\theta|$. Show $$\hat{f}(n)=\begin{cases} \pi/2& n=0 ...
1
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1answer
1k views

Inhomogenous Heat equation using fourier transform

Is it possible to transform the inhomogenous heat equation: $ u_t = u_{xx} + h(x,t)$ for $ - \infty < x< \infty , t > 0$ and $u(x,0) = 0$ to the integral equation: $$\int_0^t ...
1
vote
1answer
154 views

Fourier series and Fourier transform of a periodic function

The Fourier coefficient $C_n$ of a periodic function $s(t)$ with period $T$ is given by $$C_n= \frac{1}{T} \int_0^T s(t) e^{-2\pi int/T} \,dt$$ Now consider the Fourier term ...
0
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1answer
88 views

Upper bound for the norm of inverse Fourier tansform

Recall Hausdorff-Young inequality: For any $f\in L^p(\mathbb{R}^n)$, we have $||\hat{f}||_q\le ||f||_p$, where $p$ and $q$ are conjugate exponents and $p\in[1,2]$. It seems to me that it follows ...
0
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1answer
66 views

Bound of Fourier transfrom of a cutoff function

I am reading an article, I found that \begin{align} \mathcal{F}\chi_{[0,t_0]}(s) \end{align} is bounded by $\sqrt{1+s^2}^{-1}.$ I do not know how to get this ?
0
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1answer
82 views

Determining the amplitude of $x(t) = 3\cos^2(\omega t - \frac{\pi}{3})$ at $2\omega$ with fourier series

The given function is $x(t) = 3\cos^2(\omega t - \frac{\pi}{3})$. I have to determine the amplitude of the component with frequency $2\omega$ in the fourier series of the function. I can only do it ...
1
vote
1answer
127 views

Denoise using wavelet transform

My mathematical class task is to de-noise a function using wavelet transform. I am to select a function $f(x)$ and noise function with zero-mean $n(x)$. I am to add noise like this: $$f_{noise}(x) = ...
0
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2answers
88 views

Using fourier analysis in order to solve differential equations.

http://www.enm.bris.ac.uk/admin/courses/EMa2/Lecture%20Notes%2009-10/LSPDE5.pdf The above PDF teaches us the separation of variables method. However, there are some things I dont understand, that I ...
17
votes
1answer
292 views

Is there a combinatoric identity for the multiplicities of the following set?

Are you ready for some psychedelic pictures? Define the multiset$$S_n=\left\{\sum_{j=1}^n(-1)^{\left\lfloor(k-1)/2^{j-1}\right\rfloor}u_n^j\mbox{ for }1\leq k\leq2^n\right\}$$ where ...
3
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0answers
59 views

Period of a multivariable function

consider a function $$f(x_1, x_2, \ldots, x_n) $$ is it possible to compute the period of the function as a vector $$\langle l_1, l_2, \ldots, l_n\rangle$$ where each $l$ denotes the period of the ...
1
vote
1answer
281 views

Shifting using fouriertransform

I'm doing a small mathematical exercise where I take a function, perform a Fourier transform on it and then multiply the result by $e^{i\alpha w}$. And then take the inverse Fourier transform of the ...
3
votes
0answers
87 views

Why is the transition band of a least-square linear-phase FIR filter seems always monotonic

Given desired magnitude response and linear-phase constraint in predefined pass band and stop band, we can get the desired frequency response in both bands. By sampling the frequency in both bands, we ...
0
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1answer
141 views

What does this sentence of Greek means in the book of Modern Fourier Analysis?

I am reading Loukas Grafakos' book Modern Fourier Analysis and found this apparently Greek text in the first pages What does it mean?
1
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1answer
24 views

Is there a relation between $l_p$-norms of functions with same Fourier spectra but w.r.t different measures on the Hamming cube?

Informally, I want to ask if two functions $f$ and $g$ on the Hamming cube have the same Fourier spectra but w.r.t different measure and basis, then is $||f||_p$ related to $||g||_p$? (Where each ...
0
votes
2answers
243 views

Why is the DTFT (Discrete Time Fourier Transform) unique to each input?

As the title implies. I know the DFT of a signal is unique due to the matrix, but can anyone give a solid explanation as to why the DTFT is unique for each signal input? Thanks for your time!
0
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2answers
51 views

a differential equation equation related to fourier series

I am really struggling with this one. Any help is welcome! For equation $f''(z) + p(z) f'(z) + q(z) f(z) = 0$, where $p(z)$ and $q(z)$ are fixed polynomials. Given $f(0)=f_0$, $f'(0)=f_1$, prove that ...
2
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2answers
174 views

Understanding Dirac delta integrals?

I'm confused as to how exactly to integrate using the Dirac delta function. I have the following example: $$\int \delta (x-4)(x^3-4x^2-3x+4)dx$$ and am told this evaluates to 8. Can anyone please ...
2
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1answer
104 views

Compactly supported smooth function with Laplace transform bounded on a cone

My question is if it is possible to find a compactly supported smooth function $\varphi:\mathbf{R}\to \mathbf{R}$ s.t. the following integration $\int_{\mathbf{R}}\varphi(t)e^{itx}e^{tx}dt$ stays ...
4
votes
0answers
88 views

why is test function space $\mathcal{A}$ complete

I am trying to find out, why the space $$\mathcal{A}:=\left\{\phi\in C_0(\mathbb{R}^{2d})|\;\|\phi\|_\mathcal{A}:=\int_{\mathbb{R}^d}\sup_{x\in\mathbb{R}^d}|(\mathcal{F}_p\phi)(x,y)|\;\mathrm ...
1
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1answer
135 views

Fourier transform of a cosine function

I was reviewing a homework problem, and I'm trying to figure this out. The Fourier transform of ${1\over 2} cos(3\pi t)$, according to the solution I was given is ${1\over 2}\{\delta(f+{2\over ...
0
votes
3answers
90 views

Calculating this integral?

I'm trying to calculate $$\int\limits_{-\pi}^0e^{-x}\cos(nx)\,\mathrm{d}x$$ as part of a Fourier series calculation. My problem is the calculations seem to loop endlessly - I'm integrating by parts ...
1
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2answers
277 views

About a Fourier transform of a non- integrable function.

I'm trying to obtain the Fourier transform of the following function: $$F(x)=\frac{x}{1+x^2}$$ I have tried using Residue Theorem, but i think it can't be applied because the difference between the ...
7
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1answer
123 views

Finding a function from a fourier series

Taken from Apostol Analysis, it says, find a continuous function that generates the fourier series: $$ \sum_{n} \frac{-1^n}{n^3} \sin(nx) $$ I really have no idea how to solve this, instinctively I ...
4
votes
1answer
71 views

Fourier transform extended to $L^2$

Let $f\in L^1(\mathbb{R})\cap L^2(\mathbb{R})$, and let $f_k$ be functions in the Schwartz class such that $\|f-f_k\|_1+\|f-f_k\|_2\rightarrow 0$ as $k\rightarrow\infty$. Define ...
1
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0answers
383 views

Explicit example of a divergent Fourier series?

I've been reading Widder's Advanced Calculus text, which says that there are some continuous functions that have divergent Fourier series, which are summable to the function (C, 1). I'd greatly ...
2
votes
2answers
84 views

Prove that $f \in C^k$ if $|\hat{f}(n)|\leq C/|n|^{k+a}$

I'm looking at some problems related to Fourier series. This one stumped me a little. Suppose that $f$ is $2\pi$-periodic and piecewise smooth. Show that if there exist $k \in \mathbb{N}, a > ...
1
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0answers
85 views

DFT example in textbook

There is an example of the use of the DFT formula in my textbook which I don't quite follow. The text goes as follows: Let us define the $N$-periodic and anti-Hermitian series $g_n$ where $g_n = f_n ...
1
vote
1answer
414 views

How to convert FFT magnitude of square wave to dBm?

I wish to convert the FFT magnitude of square wave into dBm. I use FFT to covert voltage of square wave to a complex number, then i absolute the complex number into magnitude. Then i divide the ...
5
votes
1answer
260 views

Fourier transform of $f(x)=\frac{1}{e^x+e^{-x}+2}$

Let $$f(x)=\large \frac{1}{e^x+e^{-x}+2}$$ Compute the Fourier transform of $f$. We can factor the denominator to get $$f(x)=\frac1{(\exp(x/2)+\exp(-x/2))^2}=\frac1{(2\cosh(x/2))^2}$$ I'm thinking ...
2
votes
1answer
74 views

Fourier inversion formula with truncation

Let $f\in L^2(\mathbb{R})$, and denote $$s_N(x)=\dfrac{1}{2\pi}\int_{-N}^N\hat{f}(t)e^{ixt}dt.$$ Show that $$\lim_{N\rightarrow\infty}\int_\mathbb{R}|s_N(x)-f(x)|^2dx=0$$ So, $s_N(x)$ is the ...
3
votes
1answer
203 views

Fourier transform of convolution for $L^2$ functions

If $f,g\in L^1(\mathbb{R})$, it is not hard to show by definition that $$(\hat{f\ast g)}(t)=\hat{f}(t)\hat{g}(t).$$ But what about if $f,g\in L^2(\mathbb{R})$? The Fourier transform on ...
0
votes
1answer
169 views

Fourier series convergence for sum of Schwartz class functions

Let $f$ be a Schwartz class function. Let $F(x)=\sum_{n\in\mathbb{Z}}f(x-2\pi n)$. Then $F$ is periodic of period $2\pi$. How can we show that the Fourier series of $F$ converges to $F$ pointwise ...
0
votes
1answer
51 views

$f(0)$ is integral over Fourier transform for Schwartz class

Let $S$ denote the Schwartz class. Assume without proof that for every $f,g\in S$, we also have $\hat{f},\hat{g}\in S$, and $\int_\mathbb{R}f(y)\hat{g}(y)dy=\int_\mathbb{R}\hat{f}(t)g(t)dt$. Show that ...
1
vote
1answer
130 views

Wirtinger's inequality in higher dimension

Wirtinger's inequality for one-dimensional functions states that if $f(x)$, $f'(x) = \frac{df(x)}{dx}$ $\in$ $\mathcal{L}^2(a,b)$ and either $f(a) = 0$ or $f(b) = 0$ then \begin{equation} \int_{a}^{b} ...
2
votes
1answer
439 views

Use Fourier transform to calculate double integral of harmonic function

Let $$P_y(x)=\dfrac{1}{2\pi}\int_{-\infty}^\infty e^{-y|t|}e^{ixt}dt=\dfrac{1}{\pi}\dfrac{y}{x^2+y^2}.$$ Then $P_y(x)$ is harmonic in the upper half-plane $y>0$ and for $f\in L^1(\mathbb{R})$, ...
0
votes
1answer
487 views

Details for proof of Poisson summation formula

In the proof of the Poisson summation formula, there is a detail which is not clear to me how to resolve. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a Schwartz-class function. Let ...
1
vote
2answers
73 views

Calculating Fourier series of infinite sum in two ways

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be in the Schwartz class. Show that $$\sum_{n\in\mathbb{Z}}f(2\pi n)=\dfrac{1}{2\pi}\sum_{k\in\mathbb{Z}}\hat{f}(k)$$ by calculating the Fourier series of ...
1
vote
2answers
68 views

Is any periodic $C^2$ function automatically analytic?

this might be a stupid question, but is any $C^2$ function $f:\mathbb{R}\to\mathbb{C}$ of period $f(x+L)=f(x)$ automatically analytic (and in particular, infinitely often differentiable)? I learned ...
1
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1answer
117 views

About convolution and Fourier transform

I have some doubts with this question: I we have $f,g\in\cal{S}$ (where $\cal{S}$ is the Schartz space) with $f\ast g=0$, Can we deduce that $f=0$ or $g=0$? What I did is apply Fourier transform, ...
2
votes
1answer
340 views

Fourier transform and Laplace transform to solve differential equation

generally we know that both Fourier transform and Laplace transform both is used to solve differential equation,first of all let us recall both form,first Fourier transform: some times instead ...
0
votes
1answer
57 views

Complex Fourier Harmonic Oscillator

I have found the complex Fourier series for my desired force. I now need to find the steady-state forced vibration of my oscillator as a Fourier Series. (The particular solution to the inhomogeneous ...
0
votes
1answer
44 views

Continuity/Differentiability of Fourier Series

Possibly stupid question: I'm wondering if there is some trick for evaluating the continuity/differentiability of a Fourier series. In particular, I'm looking at the function $f(x)=\sum_{n=0}^\infty ...
0
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1answer
52 views

Derive Fourier transforms from Fourier expansion. How are they related?

I am just trying to relate Fourier Series expansion to Fourier Transforms. If someone could show how one value on the middle of the table is derived (from expansion) as opposed to deriving their ...
1
vote
1answer
210 views

Diagonalization of circulant matrices

Why does the following hold?: $A$ circulant matrix iff it has a representation of the form $F^{-1}DF$ where $D$ is a diagonal matrix and $F$ is a discrete Fourier transformation. I get that $F^{-1}DF$ ...
1
vote
1answer
52 views

How to pin down the complex integral?

I am working with the following problem: Find the Fourier transformation of the function $$f(x)=\frac{\sin x}{x}.$$ I did not learn any trick in complex analysis (especially various integration ...
1
vote
1answer
386 views

About the Fourier transform of the sign function

I'm trying to calculate the Fourier transform of the function $f(x):=sign(x)$. I have read some texts where this is solved approximating the function $f$ by other functions, $f_a$, defined as follows ...
3
votes
1answer
74 views

Show that for $0<t<1$, $\log\sin (t\pi)=\log(t\pi) + \sum_{n=1}^\infty\log\left(1-\frac{t^2}{n^2}\right)$

Show that for $0<t<1$, $$\log\sin (t\pi)=\log(t\pi) + \sum_{n=1}^\infty\log\left(1-\frac{t^2}{n^2}\right)$$ So I derived the following Fourier series: ...